In order to prove this estimate it is convenient to cut the manifold $M_{\epsilon}$ in various pieces and analyse the geometry separately in each of them. The global estimate follows by combining the “local” estimates obtained in each of these pieces.

Consider first the region $\rho_{j}\leq 2R_{0}\epsilon$ for some $j=1,\dots,8$. The rescaled metric $\epsilon^{-2}g_{\epsilon}$ is isometric to a compact region in the $D_{m_{j}}$ ALF space $(M_{j},g_{M_{j}})$.

Now, given a $1$–form $a$ on $M_{\epsilon}$, restrict $a$ to the region $\rho_{j}\leq 2R_{0}\epsilon$ and define $\tilde{a}=\epsilon^{-1-\delta}a$. The standard Schauder estimate for the elliptic operator $D$ associated with the metric $g_{M_{j}}$ is

Since $|\tilde{a}|_{\epsilon^{-2}g_{\epsilon}}=\epsilon^{-\delta}|a|_{g_{\epsilon}}$ and the norms $\nabla\tilde{a}$ and $D\tilde{a}$ are related in a similar way to those of $\nabla a$ and $Da$, the weighted Schauder estimate follows immediately.

The same argument can be applied in the region $\rho_{i}\leq 2R_{0}\epsilon$: the role of $M_{j}$ is now played by a small perturbation (cf. the beginning of Section 5.2) of the $A_{k_{i}-1}$ ALF space $N_{i}$.

Consider now the transition region $R_{0}\,\epsilon\leq\rho_{j}\leq\rho_{0}$ for some $j=1,\dots,8$. We can work on the double cover $H^{2m_{j}-4}$ and restrict to $\mathbb{Z}_{2}$–invariant forms. Scaling by $\epsilon$ as above we reduce to consider the restriction of $H^{2m_{j}-4}$ to the region $R_{0}\leq\rho\leq\frac{\rho_{0}}{\epsilon}$ in $\mathbb{R}^{3}$ endowed with a metric

Moreover, after rescaling the weight function $\rho_{\epsilon}$ coincides with the radial function $\rho$ on $\mathbb{R}^{3}$.

Fix a number $\sigma\in(0,1)$. For each point $x$ let $\pi(x)$ be its image in $\mathbb{R}^{3}$ and set $R=\sigma\rho(x)$. Up to changing $R_{0}$ and $\rho_{0}$ into $(1-\sigma)R_{0}$ and $(1+\sigma)\rho_{0}$ we can assume that $B_{R}(\pi(x))$ is contained in the annulus $R_{0}\leq\rho\leq\frac{\rho_{0}}{\epsilon}$ and that the restriction of $H^{2m_{j}-4}$ to this ball is trivial. We can then work on a “square” $B_{R}\times[-R,R]$ in the universal cover of $H^{2m_{j}-4}|_{B_{R}}$. Rescaling the metric by $R^{-2}$, applying standard Schauder estimates, rescaling back and multiplying by $R^{-\delta}$ we obtain

The case of the region $R_{0}\epsilon\leq\rho_{i}\leq\rho_{0}$ is completely analogous.

Finally, in the region where $\rho_{j},\rho_{i}\geq\tfrac{1}{2}\rho_{0}$ the weight function $\rho_{\epsilon}$ is uniformly equivalent to the constant $1$ and therefore weighted spaces coincide with standard Hölder spaces. Moreover the harmonic function $h_{\epsilon}$ is $C^{\infty}$–close to the constant $1$. The metric $g_{\epsilon}$ is therefore $C^{\infty}$–close to the metric $g_{\infty}=g_{\mathbb{T}}+\epsilon^{2}\theta^{2}$. The Schauder estimate for forms supported in this region is immediate since we can restrict to small balls in the torus $\mathbb{T}$ on which the circle bundle $P$ is trivial and then work on the universal cover, which has bounded geometry. ∎

By contradiction assume that there exists a sequence $\epsilon_{i}\rightarrow 0$ and $1$–forms $a_{i}$ on $M_{\epsilon_{i}}$ such that $\|a_{i}\|_{C^{1,\alpha}_{\delta}}=1$ but $\|Da_{i}\|_{C^{0,\alpha}_{\delta-1}}\rightarrow 0$.

First of all we show that for every compact set $K$ in $M^{\textup{gh}}_{\epsilon}$ which does not contain any puncture we must have $\|a_{i}\|_{C^{1,\alpha}_{\delta}(K)}\rightarrow 0$.

Over $K$ we can work on the double cover of $M^{\textup{gh}}_{\epsilon}$ and regard $a_{i}$ as $\mathbb{Z}_{2}$–invariant forms. Write $a_{i}=\epsilon_{i}f_{i}\,\theta+\gamma_{i}$, for a function $f_{i}$ and a $1$–form $\gamma_{i}$ such that $\xi\lrcorner\gamma_{i}=0$ (recall that $\xi$ is the vector field dual to $\theta$). Over $K$ we can also decompose $a_{i}=\Pi_{0}a_{i}+\Pi_{\perp}a_{i}$. Observe that for any $0<\tau<1$ (6.6) implies that

provided $\rho_{j},\rho_{i}\geq c\,\epsilon_{i}^{\frac{1-\tau}{2}}$. Since the gluing regions occur for $\rho_{j}\sim\epsilon_{i}^{\frac{2}{5}}$ and $\rho_{i}\sim\epsilon_{i}^{\frac{2}{5}}$ we can choose $\tau>0$ sufficiently small so that these assumptions are satisfied on $M^{\textup{gh}}_{\epsilon}$. Thus $\|\Pi_{\perp}a_{i}\|_{C^{0}_{\delta}(K)}\rightarrow 0$.

By the Arzelá–Ascoli Theorem we can therefore assume that $(f_{i},\gamma_{i})$ converges to $(f_{0},\gamma)\in\Omega^{0}(\mathbb{T})\oplus\Omega^{1}(\mathbb{T})$. By (6.4) $(f_{0},\gamma)$ satisfies

on $\mathbb{T}$. The control on $\|a_{i}\|_{C^{1,\alpha}_{\delta}}$ guarantees that $|f_{0}|+|\gamma|\leq C\rho^{\delta}$ close to the punctures.

We want to conclude that $f_{0}$ is constant and $\gamma$ is a smooth harmonic $1$–form on $\mathbb{T}$. By trivialising the cotangent bundle of the $3$–torus $\mathbb{T}$ by harmonic $1$–forms $\theta_{1},\theta_{2},\theta_{3}$ we can write $\gamma=f_{1}\theta_{1}+f_{2}\theta_{2}+f_{3}\theta_{3}$. Then $f_{0},f_{1},f_{2},f_{3}$ are harmonic functions on the punctured $3$–torus with controlled blow-up rate at the punctures. Since $\delta>-2$, close to each puncture we must have $f_{i}=\lambda_{i}+c_{i}\rho^{-1}+O(\rho)$ for some constants $\lambda_{i},c_{i}$. However, $c_{i}$ must vanish for all $i=0,1,2,3$ if $(f_{0},f_{i})$ is a solution of the first order system (6.12) and not only of the second order PDE this implies. Hence the functions $f_{i}$ are bounded harmonic functions on $\mathbb{T}$ and must be constant.

However, by $\mathbb{Z}_{2}$–invariance of the $1$–forms $a_{i}$ the functions $f_{i}$ must be odd with respect to the involution $\tau$ on $\mathbb{T}$ and must therefore vanish. Thus $(f_{0},\gamma)=0$ and the Schauder estimate of Proposition 6.10 implies that $\|a_{i}\|_{C^{1,\alpha}_{\delta}(K)}\rightarrow 0$.

Next we look at what happens close to one of the punctures. By what we have just proved and the assumption $\|a_{i}\|_{C^{1,\alpha}_{\delta}(M_{\epsilon})}=1$, there exists at least a $j=1,\dots,8$ or $i=1,\dots,n$ such that $\|a_{i}\|_{C^{1,\alpha}_{\delta}}\geq\tfrac{1}{n+8}>0$ in the region $\rho_{j}\leq\epsilon^{\frac{2}{5}}$ or $\rho_{i}\leq\epsilon^{\frac{2}{5}}$. We fix attention to such a region. Rescaling the metric by $\epsilon_{i}^{-2}$ and replacing $a_{i}$ with $\tilde{a}_{i}=\epsilon_{i}^{-\delta-1}a_{i}$, from now on we will work on a $D_{m}$ or an $A_{k-1}$ ALF space. Denote either of these non-compact manifolds by $M$. Because of the behaviour of the weighted Hölder norm in Definition 6.8 under rescaling, we have $\|\tilde{a}_{i}\|_{C^{1,\alpha}_{\delta}}=\|a_{i}\|_{C^{1,\alpha}_{\delta}}\geq\tfrac{1}{n+8}$. For ease of notation we replace $\tilde{a}_{i}$ with $a_{i}$ until the end of the proof.

By the Arzelá–Ascoli Theorem over every compact set of $M$ we can extract a subsequence of $\{a_{i}\}$ that converges to a solution $a$ of $Da=0$ on $M$. Moreover, $|a|\leq C\rho^{\delta}$. Since $\delta<0$ we conclude that $a=0$. Indeed, the equation $Da=0$ in particular implies that $\triangle a=0$. Since $M$ is Ricci-flat, the Weitzenböck formula yields $|a|\,\triangle|a|\leq 0$ and therefore $|a|=0$ by the maximum principle.

Now, if $\|a_{i}\|_{C^{0}_{\delta}(M)}\rightarrow 0$ then the Schauder estimate of Proposition 6.10 would yield a contradiction to the assumption $\|a_{i}\|_{C^{1,\alpha}_{\delta}}\geq\frac{1}{n+8}$. Assume therefore that there exists some $\nu>0$ such that $\|a_{i}\|_{C^{0}_{\delta}}\geq\nu$. Since $a_{i}\rightarrow 0$ in $C^{1,\alpha}_{\delta}(K)$ for every compact set $K\subset M$ there must exists a sequence of points $x_{i}\in M$ going off to infinity (in particular $R_{i}:=\rho(x_{i})\rightarrow\infty$) such that $|a_{i}(x_{i})|\geq\nu\rho(x_{i})^{\delta}$.

Now rescale the metric on $M$ by $R_{i}^{-2}$ and replace $a_{i}$ by $R_{i}^{-\delta-1}a_{i}$. Then $(M,R_{i}^{-2}g_{M})$ is converging to the tangent cone $C$ at infinity of $M$, i.e. either $C=\mathbb{R}^{3}$ or $C=\mathbb{R}^{3}/\mathbb{Z}_{2}$ depending on whether $M$ is of cyclic or dihedral type.

As in the first step of the proof, we can use (6.4) and (6.6) to conclude that $R_{i}^{-\delta-1}a_{i}$ sub-converges over compact subsets of $C\setminus\{0\}$ to a pair $(f_{0},\gamma)\in\Omega^{0}(C)\oplus\Omega^{1}(C)$ such that

$|f|+|\gamma|\leq C\rho^{\delta}$ and $(|f|+|\gamma|)(x_{0})=\nu>0$ for some $x_{0}\in C\setminus\{0\}$. As before, the fact that $\delta>-2$ implies that $f,\gamma$ are bounded close to the origin in $C$. The fact that $\delta<0$ then forces $(f_{0},\gamma)$ to vanish. However this contradicts the fact that $(|f|+|\gamma|)(x_{0})>0$. ∎